Question about logarithmic function formula

Question

I'm not good at Math and please don't blame me if this question is too obvious to you. There is a formula about logarithmic function.

$\log_ab^x = x\times\log_ab$

Here is a problem:

$\log_5x^2 = -2$

If I use the formula, I got:

$2\times log_5x=-2$

But it seems this is not correct. In the problem, $x$ could be any number including negtive numbers. After using the formula, $x$` can't be negative. So is this formula wrong or am I doing wrong somewhere?

Answer 1

You are right. In the original equation, $x$ can be negative. Writing $2\log_5x=-2$ will yield only the positive root. A better way to solve the equation is to use the definition of logarithm directly.

\begin{align*} \log_5x^2&=-2\\ x^2&=5^{-2}=\frac1{25}\\ x&=\pm\frac15 \end{align*}